On the epicomplete monoreflection of an archimedean lattice-ordered group

In a category C an object it G is epicomplete if the only epic monics out of G are isomorphisms, epic or monic meant in the categorical sense of right or left cancellable. For each of the categories Arch: archimedean ?-groups with ?-homomorphisms, and its companion category W: Arch-objects with distinguished weak unit and unit-preserving ?-homomorphisms, (and for the corresponding categories of vector lattices) epicompleteness has been characterized as divisible and conditionally and laterally σ-complete, and it has been shown to be monoreflective. Denote the reflecting functors by β and β ^W , respectively. What are they? For W the Yosida representation has been used to realize β ^W A as a certain quotient of B (Y A), the Baire functions on the Yosida space of A. For Arch, very little has been known. Here we give a general representation theorem, Theorem A, for β G as a certain subdirect product of W-epicomplete objects derived from G. That result, some W-theory, and the relation between epicity and relative uniform density are then employed to show Theorem B: β C _K (Y)=B _L (Y), where C _K (Y)is the ?-group of continuous functions on Y with compact support and B _L (Y) is the ?-group of Baire functions on Y having Lindelöf cozero sets.     

The tensor product of f-algebras

In this paper we prove that the Fremlin tensor product of two f-algebras can be endowed with an f-algebra structure and satisfies an appropriate universal property. In particular, the Riesz tensor product of C(X) and C(Y), where X and Y are topological spaces, is an f-subalgebra of C(X × Y).     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

The automorphism group of a function lattice: A problem of Jónsson and McKenzie

It is shown that Aut(L ^Q ) is naturally isomorphic to Aut(L) × Aut(Q) whenL is a directly and exponentially indecomposable lattice,Q a non-empty connected poset, and one of the following holds:Q is arbitrary butL is ajm-lattice,Q is finitely factorable and L is complete with a join-dense subset of completely join-irreducible elements, orL is arbitrary butQ is finite. A problem of Jónsson and McKenzie is thereby solved. Sharp conditions are found guaranteeing the injectivity of the natural mapv _P,Q from Aut(P) × Aut(Q) to Aut(P ^Q )P andQ posets), correcting misstatements made by previous authors. It is proven that, for a bounded posetP and arbitraryQ, the Dedekind-MacNeille completion ofP ^Q ,DM(P ^Q ), is isomorphic toDM(P)^Q. This isomorphism is used to prove that the natural mapv _P,Q is an isomorphism ifv _DM(P),Q is, reducing a poset problem to a more tractable lattice problem.Presented by B. Jonsson.The author would like to thank his supervisor, Dr. H. A. Priestley, for her direction and advice as well as his undergraduate supervisor, Prof. Garrett Birkhoff, and Dr. P. M. Neumann for comments regarding the paper. This material is based upon work supported under a (U.S.) National Science Foundation Graduate Research Fellowship and a Marshall Aid Commemoration Commission Scholarship.     

K0 of multiplier algebras of C*-algebras with real rank zero

For a large class of -unital C ^*-algebras A with real rank zero and stable rank one, the structure of the Grothendieck group k ₀ of the multiplier algebra (A) is investigated. The ordered group K ₀( (A)) is shown to be an unperforated Riesz group, and its additive structure is completely determined, as is — in important cases — its order structure. These structures, and the attendant consequences for the ideal structure of (A), are richer than previously anticipated. In particular, it is shown that the corona algebra (A)/A can have very large stably finite quotient algebras. For example, there exist simple, separable, approximately finite-dimensional C ^*-algebras A such that the maximal stably finite quotient algebra of (A)/A has uncountably many maximal ideals modulo which a W ^*-factor of Type II₁ results. The analysis of the additive structure of K ₀( (A)) yields as a byproduct that if A is a -unital approximately finite-dimensional C ^*-algebra without nonzero unital quotient algebras, then all quasitraces on (A) are traces.This research was partially supported by a grant from the National Science Foundation.     

Lattice-ordered groups whose lattices of convex l-subgroups guarantee noncommutativity

Paul Conrad has asked whether for every lattice-ordered group G, there exists an Abelian N sharing its lattice C(G) of convex l-subgroups, i.e., such that C(N)C(G). Several counter-examples are exhibited, including all free l-groups of uncountably infinite rank. Also, for A() the l-group of order-automorphisms of the real line , it is shown that any N sharing its C would have to be an l-subgroup of the l-group C() of continuous real-valued functions-and that C(C())C(A()).This work was done while the author was spending a most enjoyable year at Boise State University.     

C-semigroups and strongly continuous semigroups

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| _Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖_[Im(C)]≡‖C ⁻¹ x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| _X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.     

On the Structure of the Completion of a Normed Lattice

Let X be a normed lattice and Y be the norm completion of X with a natural embedding π : X → Y . By the Kawai- Luxemburg theorem, X is embedded as an order dense set and π preserves all suprema and infima iff X satisfies the condition (A_o ) (i.e., the norm has pseudo σ-Lebesgue property). Let X_o be the largest ideal in X having the condition (A_o); let Y_(o) be the band in Y generated by πX_o and Y₍₁₎ be the complementary band to Y_(o). The structure of Y and, in particular, of the bands Y_(o) and Y₍₁₎ are studied. The conditions for Y_(o) to be a projection band and πX_o to be topologically dense in Y_(o) are obtained.     

ON Subspaces OF NON-Reflexive Orlicz Spaces

Abstract

Kadec and Pelczýnski have shown that every non-reflexive subspace of L ¹ (μ) contains a copy of l ₁ complemented in L ¹(μ). On the other hand Rosenthal investigated the structure of reflexive subspaces of L ¹(μ) and proved that such sub-spaces have non-trivial type. We show the same facts to hold true for a special class of non-reflexive Orlicz spaces. In particular we show that if F is an N-function in δ₂ with its complement G satisfying limt→∞ G(ct)/G(t) = ∞, then every non-reflexive subspace of L*_F contains a copy of l ₁ complemented in L*_F. Furthermore we establish the fact that if F is an N-function in δ₂ with its complement G satisfying limt→∞ G(ct)/G(t) = ∞, then every reflexive subspace of L*_F has non-trivial type.     

Remarks on intermediate C-rings of C(X)

Let X be a Tychono? space and A(X) be a subring of C(X) containing C^?(X). We introduce the notion of -ideal in A(X). It is observed that the class of -ideals contains the class of z_A-ideals and is contained in the class of z-ideals of A(X). These containments may be proper. It turns out that coincidence of z-ideals of A(X) with -ideals characterizes intermediate C-rings of C(X).     

Schauder type theorems for differentiable and holomorphic mappings

Denoting byC _wu ^p (E) the algebra of allC ^p-real-valued functions on the real Banach spaceE such that the functions and the derivatives are weakly uniformly continuous on bounded subsets, it is known that the algebra homomorphismsA:C _wu ^q (F)C _wu ^p (E) are induced by differentiable mappingsg:EF ^**. We prove that, for 1p+1q, the following are equivalent: (a)A is compact; (b)g and its derivatives are compact; (c)gC _wu ^p (E,F ^**) (the authors had proved that, forp=q<,A is [weakly] compact if and only ifg is a constant mapping, and it is known that ifq<p, thenA is always induced by a constant mapping and is therefore compact). Also, for an entire function of bounded typegH _b (U,F), where is a balanced open subset, andE,F are complex Banach spaces, lettingA:H _b (F)H _b (U) be the homomorphism given byA(f)=fg for allfH _b (F), we prove thatA is compact if and only ifg is compact.Supported in part by DGICYT Grant PB 94-1052 (Spain).Supported in part by DGICYT Grant PB 93-0452 (Spain).     

Zero-separating algebras of continuous functions

Let A be a lattice-ordered algebra endowed with a topology compatible with the structure of algebra. We provide internal conditions for A to be isomorphic as lattice-ordered algebras and homeomorphic to Ck(X), the lattice-ordered algebra C(X) of real continuous functions on a completely regular and Hausdorff topological space X, endowed with the topology of uniform convergence on compact sets. As a previous step, we determine this topology among the locally m-convex topologies on C(X) with the property that each order closed interval is bounded.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Copies of c0(Г) and ?∞(Г)/c0(Г) in quotients of Banach spaces with applications to Orlicz and Marcinkiewicz spaces

Let X be a Banach space, let Y be its subspace, and let Г be an infinite set. We study the consequences of the assumption that an operator T embeds ?_221E;(Г) into X isomorphically with T(c₀(Г)) ⊂ Y. Under additional assumptions on T we prove the existence of isomorphic copies of c₀(Г^ℵ0) in X/Y, and complemented copies ?_∞(Г) in X/Y. In concrete cases we obtain a new information about the structure of X/Y. In particular, L∞[O,1]/C[O,1] contains a complemented copy of ?_∞/c₀, and some natural (lattice) quotients of real Orlicz and Marcinkiewicz spaces contain lattice-isometric and positively I-complemented copies of(real) ?_∞/c₀.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

LINEAR MAPPINGS THAT PRESERVE THE DERIVATIONAL STRUCTURE OF C*-ALGEBRAS

Abstract

Given a C*-algebra A and a suitable set of derivations on A, we consider the algebras A _n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C^(m)(U) → C ^(p) (V) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). As part of the theory developed we obtain a non-trivial extension of the Kleinecke-Shirokov theorem in the category of C*-algebras to unbounded partially defined *-derivations. This indicates the existence of a single mathematical principle governing both the non-increasibility of differentiability by continuous homomorphisms and the untenability of the Heisenberg Uncertainty Principle for bounded observables.     

Weakly Singular Integral Operators in Weighted L∞–Spaces

We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N₁, t ∈N₂, or x=t, where N₁,N₂ are sets of measure zero. It is shown that such operators map weighted L^∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors of best weighted uniform approximation by algebraic polynomials.